We consider a class of queueing systems that consist of server pools in parallel and multiple customer classes. Customer service times are assumed to be exponentially distributed. We study the asymptotic behavior of these queueing systems in a heavy traffic regime that is known as the Halfin and Whitt many-server asymptotic regime. Our main contribution is a general framework for establishing state space collapse results in the Halfin and Whitt many-server asymptotic regime for parallel server systems having multiple customer classes. In our work, state space collapse refers to a decrease in the dimension of the processes tracking the number of customers in each class waiting for service and the number of customers in each class being served by various server pools. We define and introduce a state space collapse function, which governs the exact details of the state space collapse. Our methodology is similar in spirit to that in Bramson (1998); however, Bramson studies an asymptotic regime in which the number of servers is fixed and Bramson does not require a state space collapse function. We illustrate the applications of our results in three different parallel server systems. The first system is a distributed parallel server system under the minimum-expected-delay faster-server-first (MED-FSF) or minimumexpected- delay load-balancing (MED-LB) policies. We prove that the MED-FSF policy minimizes the stationary distribution of total number of customers in the system. However, under the MED-FSF policy all the servers in our distributed system except those with the lowest service rate experience 100% utilization but under the MED-LB policy, on the other hand, the utilizations of all the server pools are equal. The second system we consider is known as the N-model. We show that when the service times only depend on the server pool providing service a static priority rule is asymptotically optimal. Finally, we study two results conjectured in the literature for V-systems. We show for all of these systems that the conditions on the hydrodynamic limits can easily be checked using the standard tools that have been developed in the literature to analyze fluid models.

Wireless technology has become an increasingly popular way to gain network access. Wireless networks are expected to provide efficient and reliable service and support a broad range of emerging applications, such as multimedia streaming and video conferencing. However, limited wireless spectrum together with interference and fading pose signi cant challenges for network designers. The novel technique of network coding has a significant potential for improving the throughput and reliability of wireless networks by taking advantage of the broadcast nature of wireless medium. Reverse carpooling is one of the main techniques used to realize the benefits of network coding in wireless networks. With reverse carpooling, two flows are traveling in opposite directions, sharing a common path. The network coding is performed in the intermediate (relay) nodes, which saves up to 50% of transmissions. In this thesis, we focus on the scheduling at the relay nodes in wireless networks with reverse carpooling. When two packets traveling in opposite directions are available at the relay node, the relay node combines them and broadcasts the resulting packet. This event is referred to as a coding opportunity. When only one packet is available, the relay node needs to decide whether to wait for future coding opportunities, or to transmit them without coding. Though the choice of holding packets exploits the positive aspects of network coding, without a proper policy in place that controls how long the packets should wait, it will have an adverse impact on delays and thus the overall network performance. Accordingly, our goal is to find an optimal control strategy that delicately balances the tradeoff between the number of transmissions and delays incurred by the packets. We also address the fundamental question of what local information we should keep track of and use in making the decision of of whether to transmit uncoded packet or wait for the next coding opportunity. The available information consists of queue length and time stamps indicating the arrival time of packets in the queue. We could also store history of all previous states and actions. However, using all this information makes the control very complex and so we try to find if the overhead in collecting waiting times and historical information is worth it. A major contribution of this thesis is a stochastic control framework that uses state information based on what can be observed and prescribes an optimal action. For that, we formulate and solve a stochastic dynamic program with the objective of minimizing the long run average cost per unit time incurred due to transmissions and delays. Subsequently, we show that a stationary policy based on queue lengths is optimal, and the optimal policy is of threshold-type. Then, we describe a non-linear optimization procedure to obtain the optimal thresholds. Further, we substantiate our analytical ndings by performing numerical experiments under varied settings. We compare systems that use only queue length with those where more information is available, and we show that optimal control that uses only the queue length is as good as any optimal control that relies on knowing the entire history.

Transportation is one of the most vital services in modern society. It makes most of the other functions of society possible. Real transportation systems are so large and complex that in order to build the science of transportation systems it will be necessary to work in many areas, such as: Modeling, Optimization and Simulation. We are interested in solutions for the so-called fleet-sizing-and-allocation problem (FSAP). Fleet sizing and allocation problems are one of the most interesting and hard to solve logistic problems. A fleet sizing and allocation problem consists of two interdependent parts. The fleet sizing problem is to determine a number of transportation units that optimally balances service requirements against the cost of purchasing and maintaining the transportation units. The allocation problem is dealing with the repositioning of transportation units to serve future transportation demand. To make the fleet sizing and allocation problem a little bit more tractable we concentrate on logistic systems with a special hub-and-spoke structure. We start with a very simple fleet sizing of one-to-one case. This case will cause us to focus attention on several key issues in fleet sizing. Afterwards, the generalization of the one-to-one system is the one-to-many system. As a simple example can serve the continuous time situation where a single origin delivers items to many destinations. For the case that items are produced in a deterministic production cycle and transportation times are stochastic. We also studied a hub-and-spoke problem with continuous time and stochastic demand. To solve this problem, based on Marginal Analysis, we applied queueing theory methods. The investigation of the fleet-sizing-and-allocation problem for hub-and-spoke systems is started for a single-period, deterministic-demand model. In that the model hub has to decide how to use a given number of TU’s to satisfy a known (deterministic) demand in the spokes. We consider two cases: 1. Renting of additional TU’s from outside the system is not possible, 2. Renting of additional TU’s from outside the system is possible. For each case, based on Marginal Analysis, we developed a simple algorithm, which gives us the cost-minimal allocation. Since the multi-period, deterministic demand problem is NP-hard we suggest to use Genetic Algorithms. Some building elements for these are described. For the most general situation we also suggest to use simulation optimization. To realize the simulation optimization approach we could use the software tool “Calculation Assessment Optimization System” (CAOS). The idea of CAOS is to provide a software system, which separates the optimization process from the optimization problem. To solve an optimization problem the user of CAOS has to build up a model of the system to which the problem is related. Furthermore he has to define the decision parameters and their domain. Finally, we used CAOS for two classes of hub-and-spoke system: 1. A single hub with four spokes, 2. A single hub with fifty spokes. We applied four optimizers – a Genetic Algorithm, Tabu Search, Hybrid Parallel and Hybrid Serial with two distributions (Normal Distribution and Exponential Distribution) for a customer interarrival times and their demand.

Modelling is an important technique in the comprehension and management of complex systems. Queueing network models capture most relevant information from computer system and network behaviour. The construction and resolution of these models is constrained by many factors. Approximations contain detail lost for exact solution and/or provide results at lower cost than simulation. Information at the resource and interactive command level is gathered with monitors under ULTRIX'. Validation studies indicate central processor service times are highly variable on the system. More pessimistic predictions assuming this variability are in part verified by observation. The utility of the Generalised Exponential (GE) as a distribution parameterised by mean and variance is explored. Small networks of GE service centres can be solved exactly using methods proposed for Generalised Stochastic Petri Nets. For two centre. systems of GE type a new technique simplifying the balance equations is developed. A very efficient "building bglloocbka"l. is presented for exactly solving two centre systems with service or transfer blocking, Bernoulli feedback and load dependent rate, multiple GE servers. In the tandem finite buffer algorithm the building block illustrates problems encountered modelling high variability in blocking networks. A parametric validation study is made of approximations for single class closed networks of First-Come-First-Served (FCFS) centres with general service times. The multiserver extension using the building block is validated. Finally the Maximum Entropy approximation is extended to FCFS centres with multiple chains and implemented with computationally efficient convolution.

Traditionally the Internet is used for the following applications: FTP, e-mail and Web traffic. However in the recent years the Internet is increasingly supporting emerging applications such as IP telephony, video conferencing and online games. These new applications have different requirements in terms of throughput and delay than traditional applications. For example, interactive multimedia applications, unlike traditional applications, have more strict delay constraints and less strict loss constraints. Unfortunately, the current Internet offers only a best-effort service to all applications without any consideration to the applications specific requirements. In this thesis three existing Active Queue Management (AQM) mechanisms are modified by incorporating into these a control function to condition routers for better Quality of Service (QoS). Specifically, delay is considered as the key QoS metric as it is the most important metric for real-time multimedia applications. The first modified mechanism is Drop Tail (DT), which is a simple mechanism in comparison with most AQM schemes. A dynamic threshold has been added to DT in order to maintain packet queueing delay at a specified value. The modified mechanism is referred to as Adaptive Drop Tail (ADT). The second mechanism considered is Early Random Drop (ERD) and, iii in a similar way to ADT, a dynamic threshold has been used to keep the delay at a required value, the main difference being that packets are now dropped probabilistically before the queue reaches full capacity. This mechanism is referred to as Adaptive Early Random Drop (AERD). The final mechanism considered is motivated by the well known Random Early Detection AQM mechanism and is effectively a multi-threshold version of AERD in which packets are dropped with a linear function between the two thresholds and the second threshold is moveable in order to change the slope of the dropping function. This mechanism is called Multi Threshold Adaptive Early Random Drop (MTAERD) and is used in a similar way to the other mechanisms to maintain delay around a specified level. The main focus with all the mechanisms is on queueing delay, which is a significant component of end-to-end delay, and also on reducing the jitter (delay variation) A control algorithm is developed using an analytical model that specifies the delay as a function of the queue threshold position and this function has been used in a simulation to adjust the threshold to an effective value to maintain the delay around a specified value as the packet arrival rate changes over time. iv A two state Markov Modulated Poisson Process is used as the arrival process to each of the three systems to introduce burstiness and correlation of the packet inter-arrival times and to present sudden changes in the arrival process as might be encountered when TCP is used as the transport protocol and step changes the size of its congestion window. In the investigations it is assumed the traffic source is a mixture of TCP and UDP traffic and that the mechanisms conserved apply to the TCP based data. It is also assumed that this consists of the majority proportion of the total traffic so that the control mechanisms have a significant effect on controlling the overall delay. The three mechanisms are evaluated using a Java framework and results are presented showing the amount of improvement in QoS that can be achieved by the mechanisms over their non-adaptive counterparts. The mechanisms are also compared with each other and conclusions drawn.

The central thesis of this dissertation is that by combining classical scheduling methodologies with those of inventory management and queueing theory we can better model, understand and solve complex real-world scheduling problems. In part II of this dissertation, we provide models of a realistic supply chain scheduling problem that capture both its combinatorial nature and its dependence on inventory availability. We present an extensive empirical evaluation of how well implementations of these models in commercially available software solve the problem. We are therefore able to address, within a specific problem, the need for scheduling to take into account related decision-making processes. In order to simultaneously deal with combinatorial and dynamic properties of real scheduling problems, in part III we propose to integrate queueing theory and deterministic scheduling. Firstly, by reviewing the queueing theory literature that deals with dynamic resource allocation and sequencing and outlining numerous future work directions, we build a strong foundation for the investigation of the integration of queueing theory and scheduling. Subsequently, we demonstrate that integration can take place on three levels: conceptual, theoretical and algorithmic. At the conceptual level, we combine concepts, ideas and problem settings from the two areas, showing that such combinations provide insights into the trade-off between long-run and short-run objectives. Next, we show that theoretical integration of queueing and scheduling can lead to long-run performance guarantees for scheduling algorithms that have previously been proved only for queueing policies. In particular, we are the first to prove, in two flow shop environments, the stability of a scheduling method that is based on the traditional scheduling literature and utilizes processing time information to make sequencing decisions. Finally, to address the algorithmic level of integration, we present, in an extensive future work chapter, one general approach for creating hybrid queueing/scheduling algorithms. To our knowledge, this dissertation is the first work that builds a framework for integrating queueing theory and scheduling. Motivated by characteristics of real problems, this dissertation takes a step toward extending scheduling research beyond traditional assumptions and addressing more realistic scheduling problems.

The central thesis of this dissertation is that by combining classical scheduling methodologies with those of inventory management and queueing theory we can better model, understand and solve complex real-world scheduling problems. In part II of this dissertation, we provide models of a realistic supply chain scheduling problem that capture both its combinatorial nature and its dependence on inventory availability. We present an extensive empirical evaluation of how well implementations of these models in commercially available software solve the problem. We are therefore able to address, within a specific problem, the need for scheduling to take into account related decision-making processes. In order to simultaneously deal with combinatorial and dynamic properties of real scheduling problems, in part III we propose to integrate queueing theory and deterministic scheduling. Firstly, by reviewing the queueing theory literature that deals with dynamic resource allocation and sequencing and outlining numerous future work directions, we build a strong foundation for the investigation of the integration of queueing theory and scheduling. Subsequently, we demonstrate that integration can take place on three levels: conceptual, theoretical and algorithmic. At the conceptual level, we combine concepts, ideas and problem settings from the two areas, showing that such combinations provide insights into the trade-off between long-run and short-run objectives. Next, we show that theoretical integration of queueing and scheduling can lead to long-run performance guarantees for scheduling algorithms that have previously been proved only for queueing policies. In particular, we are the first to prove, in two flow shop environments, the stability of a scheduling method that is based on the traditional scheduling literature and utilizes processing time information to make sequencing decisions. Finally, to address the algorithmic level of integration, we present, in an extensive future work chapter, one general approach for creating hybrid queueing/scheduling algorithms. To our knowledge, this dissertation is the first work that builds a framework for integrating queueing theory and scheduling. Motivated by characteristics of real problems, this dissertation takes a step toward extending scheduling research beyond traditional assumptions and addressing more realistic scheduling problems.

This thesis focuses on increasing the efficiency of systems with cross-trained workforce and finite storage spaces. Our objective is to maximize the throughput and minimize the setup costs (if they exist). More specifically, we are interested in determining effective cross-training strategies and dynamic server assignment policies for flexible servers in production lines with finite buffers. In the first part of this thesis, we study non-Markovian systems and support the conjecture that effective server assignment policies are robust to service time distributions. Next, we consider understaffed tandem lines with partially or fully flexible servers, determine optimal and heuristic server assignment policies, and show that most of the benefits of full flexibility can be achieved with limited flexibility. Finally, we incorporate the setups to our model, determine the optimal server assignment policy for some systems and show how the effective assignment of servers depends on the magnitude of the setup costs.

In the past few years, we have witnessed the continuous growth in popularity of delay-sensitive applications. Applications like live video streaming, multimedia conferencing, VoIP and online gaming account for a major part of Internet traffic these days. It is also predicted that this trend will continue in the coming years. This emphasizes the significance of developing efficient scheduling algorithms in communication networks with guaranteed low delay performance. In our work, we try to address the delay issue in some major instances of wireless communication networks. First, we study a wireless content distribution network (CDN), in which the requests for the content may have service deadlines. Our wireless CDN consists of a media vault that hosts all the content in the system and a number of local servers (base stations), each having a cache for temporarily storing a subset of the content. There are two major questions associated with this framework: (i) content caching: which content should be loaded in each cache? and (ii) wireless network scheduling: how to appropriately schedule the transmissions from wireless servers? Using ideas from queuing theory, we develop provably optimal algorithms to jointly solve the caching and scheduling problems. Next, we focus on wireless relay networks. It is well accepted that network coding can enhance the performance of these networks by exploiting the broadcast nature of the wireless medium. This improvement is usually evaluated in terms of the number of required transmissions for delivering flow packets to their destinations. In this work, we study the effect of delay on the performance of network coding by characterizing a trade-off between latency and the performance gain achieved by employing network coding. More specifically, we associate a holding cost for delaying packets before delivery and a transmission cost for each broadcast transmission made by the relay node. Using a Markov decision process (MDP) argument, we prove a simple threshold-based policy is optimal in the sense of minimum long-run average cost. Finally, we analyze delay-sensitive applications in wireless peer-to-peer (P2P) networks. We consider a hybrid network which consists of (i) an expensive base station-to-peer (B2P) network with unicast transmissions, and (ii) a free broadcast P2P network. In such a framework, we study two popular applications: (a) a content distribution application with service deadlines, and (b) a multimedia live streaming application. In both problems, we utilize random linear network coding over finite fields to simplify the coordination of the transmissions. For these applications, we provide efficient algorithms to schedule the transmissions such that some quality of service (QoS) requirements are satisfied with the minimum cost of B2P usage. The algorithms are proven to be throughput optimal for sufficiently large field sizes and perform reasonably well for finite fields.

Σκοπός της παρούσας διπλωματικής εργασίας είναι η μελέτη της θεωρίας της Near Complete Decomposability (NCD) και η εφαρμογή της στην ανάλυση της απόδοσης ενός υπολογιστικού συστήματος, του οποίου η μοντελοποίηση με παραδοσιακές τεχνικές οδηγεί σε απαγορευτικά μεγάλο χώρο κατάστασης. Αρχικά, παραθέτουμε τα βασικά σημεία της θεωρίας όπως αυτή θεμελιώνεται μαθηματικά από τον Courtois στην κλασική του μονογραφία (Courtois, 1977), ενώ στη συνέχεια προβαίνουμε στη μοντελοποίηση ενός υποθετικού σταθμού εργασίας κάποιου πολυεπεξεργαστικού συστήματος, στο οποίο εκτελούνται ανά πάσα στιγμή το πολύ Κ έργα. Ο σταθμός εργασίας που μελετάμε διαθέτει buffer πεπερασμένου μεγέθους και είναι επιφορτισμένος με τη συγκέντρωση και το συνδυασμό των επιμέρους υποέργων κάθε έργου και την αποθήκευση του στη μνήμη. Οι κλασικές τεχνικές μοντελοποίησης του buffer οδηγούν σε ένα μοντέλο με πολύ μεγάλο χώρο κατάστασης. Ωστόσο εμείς μοντελοποιούμε μία συναθροιστική εκδοχή του αρχικού μοντέλου, η οποία υπό αρκετά ρεαλιστικές συνθήκες χαίρει της NCD ιδιότητας. Την ιδιότητα αυτή του μοντέλου μας τη δικαιολογούμε τόσο διαισθητικά, όσο και μαθηματικά. Επίσης, επιβεβαιώνουμε πως το NCD μοντέλο πετυχαίνει υψηλής ποιότητας εκτίμηση των πιθανοτήτων μόνιμης κατάστασης και μίας σειράς άλλων χρήσιμων μετρικών, με σημαντικά μικρότερο υπολογιστικό κόστος σε σχέση με το αρχικό μοντέλο, εκτελώντας μία σειρά μετρήσεων στο περιβάλλον Matlab. Παράλληλα, η αξιοποίηση του NCD μοντέλου αυξάνει σημαντικά την ικανότητά μας να ερμηνεύσουμε τη δυναμική συμπεριφορά του συστήματος καθώς αυτό οδεύει προς μια κατάσταση στατιστικής ισορροπίας. Τέλος, επιχειρούμε μία σειρά από “educated guesses” για πιθανές κλάσεις συστημάτων τα οποία θα μπορούσαν να αναλυθούν με μεθοδολογία αντίστοιχη με αυτήν που ακολουθήσαμε εμείς στο παρόν κείμενο. / The purpose of this diploma dissertation is, on one hand the brief study of the theory of Near Complete Decomposability (NCD), and on the other hand the application of NCD in the analysis of a system, the modeling of which leads to a prohibitively large state space. First, we point out the fundamental mathematical principles of NCD as established by Courtois in his classic monograph (Courtois, 1977). Then, we proceed to the modeling of a hypothetical service station (R) of a multiprocessing computer system, which executes at most K jobs simultaneously. R has a finite buffer and its duty is to combine the arriving tasks into a single job and store it to memory. The usual modeling techniques applied to this “task buffer”, lead to a model with extremely large state space. So, we construct a lumped model instead, which enjoys the property of NCD. We prove this, using intuitive arguments as well as mathematical ones. Then, we confirm that the NCD model achieves a reliable estimation of the steady state probability vector and other important metrics, with significantly reduced computational complexity in comparison with the initial model. Furthermore, the exploitation of the NCD model increases significantly our ability to understand the dynamics of our system and to interpret aspects of its transient behavior towards statistical equilibrium. Finally, we make a number of “educated guesses” about possible classes of systems that could be analyzed using the same kind of techniques we used in this dissertation.